Transductive Conformal Inference With Adaptive Scores: Exact Formulas for Pn,m

Table of Links

• Abstract & Introduction
• Main results
• Application to prediction intervals
• Application to novelty detection
• Conclusion, Acknowledgements and References
• Appendix A: Exact formulas for Pn,m
• Appendix B: Numerical bounds and templates
• Appendix C: Proof
• Appendix D: Explicit control of (16) for α=0
• Appendix E: Proof of Corollary 4.1
• Appendix F: The Simes inequality
• Appendix G: Uniform FDP bound for AdaDetect
• Appendix H: Additional experiments

A Exact formulas for Pn,m

Theorem A.1 (ii) provides the exact dependency structure between the p-values: for instance, M(j)! = 1 when the coordinates of j = (j1, . . . , jm) are all distinct, while M(j)! = m! when the coordinates of j = (j1, . . . , jm) are the same. This means that the distribution slightly favors the j with repeated entries. This shows that the conformal p-values are not i.i.d. but have a positive structure of dependency. This is in accordance with the specific positive dependence property (called PRDS) already shown by Bates et al. (2023); Marandon et al. (2022).